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Cost, revenue, and profit are in dollars and x is the number of units. Suppose that the marginal revenue for a product is MR = 3600 and the marginal cost is MC = 120 sqrt(x + 4) with a fixed cost of $600. (a) Find the profit or loss from the production and sale of 5 units. (b) How many units will result in a maximum profit?

Guest Apr 30, 2017
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I'll give it a go but it might be wrong.  I'm pretty confident on part b but less confident with part a.

 

Cost, revenue, and profit are in dollars and x is the number of units.

Suppose that the marginal revenue for a product is MR = 3600 and the marginal cost is MC = 120 sqrt(x + 4) with a fixed cost of $600.

 

(a) Find the profit or loss from the production and sale of 5 units.

 

Revenue for producing and selling 5 units = 3600*5 = $18000

 

\(\text{MC for 1st unit }= 120\sqrt{5}\\ \text{MC for 2nd unit }= 120\sqrt{6}\\ \text{MC for 3rd unit }= 120\sqrt{7}\\ \text{MC for 4th unit }= 120\sqrt{8}\\ \text{MC for 5th unit }= 120\sqrt{9}\\\)

Sum of marginal costs = 120*(sqr5+sqrt6+sqrt7+sqrt8+3) = $1579

marginal costs+FC = 1579+600 = $2179

Profit on 5 units = 18000-2179 = $15821

 

(b) How many units will result in a maximum profit?

 

\(\text{MProfit = marginal revenues - marginal costs} \\ MP=3600-120\sqrt{x+4}\\ \text{Profit is maximised when marginal profit =0}\\ 3600-120\sqrt{x+4}=0\\ 30-\sqrt{x+4}=0\\ \sqrt{x+4}=30\\ x+4=900\\ x=896\;units \)

Melody  May 1, 2017

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